Models of Language EvolutionEvolutionary game theory & signaling games

Michael Franke

Topics for today

1 (flavors of) game theory

2 signaling games (& conversion into symmetric form)

3 Nash equilibrium (in symmetric games)

4 evolutionary stability

5 meaning of signals

Game Theory Signaling games Population Games

Game Theory

Signaling games

Population Games

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Game Theory Signaling games Population Games

(Rational) Choice Theory

Decision Theory: a single agent’s solitary decision

Game Theory: multiple agents’ interactive decision making

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Game Theory Signaling games Population Games

Game Theory

• abstract mathematical tools for modeling and analyzing multi-agent interaction• since 1940: classical game theory (von Neumann and Morgenstern)

• perfectly rational agents ::: Nash equilibrium• initially promised to be a unifying formal foundation for all social sciences• Nobel laureates: Nash, Harsanyi & Selten (1994), Aumann & Schelling (2006)

• since 1970: evolutionary game theory (Maynard-Smith, Prize)• boundedly-rational agents ::: evolutionary stability & replicator dynamics• first applications in biology, later also elsewhere (linguistics, philosophy)

• since 1990: behavioral game theory (Selten, Camerer)• studies interactive decision making in the lab

• since 1990: epistemic game theory (Harsanyi, Aumann)• studies which (rational) beliefs of agents support which solution concepts

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Game Theory Signaling games Population Games

Games vs. Behavior

Game: abstract model of a recurring interactive decision situation• think: a model of the environment

Strategies: all possible ways of playing the game• think: a full contingency plan or a (biological) predisposition for how to act in every

possible situation in the game

Solution: subset of “good strategies” for a given game• think: strategies that are in equilibrium, rational, evolutionarily stable, the outcome of

some underlying agent-based optimization process etc.

Solution concept: a general mapping from any game to its specific solution• examples: Nash equilibrium, evolutionary stability, rationalizability etc.

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Game Theory Signaling games Population Games

Kinds of Games

uncertainty choice points

simultaneous in sequence

no strategic/static dynamic/sequentialwith complete info

yes Bayesian dynamic/sequentialwith incomplete info

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Game Theory Signaling games Population Games

Game Theory

Signaling games

Population Games

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Game Theory Signaling games Population Games

t ∈ T

?

sender knows state, but receiver does not

t ∈ Tm ∈ M

sender sends a signal

a ∈ A

receiver chooses act

State-Act Payoff Matrixa1 a2 . . .

t1 1,1 0,0t2 1,0 0,1...

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(Lewis, 1969)

Game Theory Signaling games Population Games

Signaling game

A signaling game is a tuple

〈{S, R} , T, Pr, M, A, US, UR〉with:

{S, R} set of players

T set of states

Pr prior beliefs: Pr ∈ ∆(T)

M set of messages

A set of receiver actions

US,R utility functions:T×M×A→ R .

Talk is cheap iff for all t, m, m′, a andX ∈ {S, R}:

UX(t, m, a) = UX(t, m′, a) .

Otherwise we speak of costly signaling.

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model of the context/environment/world

Game Theory Signaling games Population Games

Example (2-2-2 Lewis game)2 states, 2 messages, 2 acts

Pr(t) a1 a2

t1 p 1, 1 0, 0

t2 1− p 0, 0 1, 1

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Game Theory Signaling games Population Games

Example (Alarm calls)

NS S

R R

R R

〈1, 1〉 〈0, 0〉

〈1, 1〉 〈0, 0〉

〈0, 0〉 〈1, 1〉

〈0, 0〉 〈1, 1〉

pt1

1− pt2

m2

m1

m2

m1

a1 a2 a1 a2

a1 a2 a1 a2

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Game Theory Signaling games Population Games

Strategies

Pure

s ∈ MT r ∈ AM fixed contingency plan

Mixed

s̃ ∈ ∆(MT) r̃ ∈ ∆(AM) uncertainty about plan

Behavioral

σ ∈ (∆(M))T ρ ∈ (∆(A))M probabilistic plan

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Game Theory Signaling games Population Games

Pure sender strategies in the 2-2-2 Lewis game

“mamb”:ma

mb

t1

t2

“mbma”:ma

mb

t1

t2

“mama”:

ma

mb

t1

t2

“mbmb”:ma

mb

t1

t2

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Game Theory Signaling games Population Games

Pure receiver strategies in the 2-2-2 Lewis game

“aaab”:a1

a2

ma

mb

“abaa”:a1

a2

ma

mb

“aaaa”:

a1

a2

ma

mb

“abab”:a1

a2

ma

mb

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Game Theory Signaling games Population Games

All pairs of sender-receiver pure strategies for the 2-2-2 Lewis game

13

9

5

1

14

10

6

2

15

11

7

3

16

12

8

4

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Game Theory Signaling games Population Games

Game Theory

Signaling games

Population Games

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Game Theory Signaling games Population Games

(One-Population) Symmetric Game

A (one-population) symmetric game is a pair 〈A, U〉, where:• A is a set of acts, and• U : A×A→ R is a utility function (matrix).

Example (Prisoner’s dilemma)

U =

( ac ad

ac 2 0

ad 3 1

)Example (Hawk & Dove)

U =

( ah ad

ah 1 7

ad 2 3

)

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Game Theory Signaling games Population Games

Mixed strategies in symmetric games

A mixed strategy in a symmetric game is a probability distribution σ ∈ ∆(A).

Utility of mixed strategies defined as usual:

U(σ, σ′) = ∑a,a′∈A

σ(a)× σ(a′)×U(a, a′)

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Game Theory Signaling games Population Games

Nash Equilibrium in Symmetric Games

A mixed strategy σ ∈ ∆(A) is a symmetric Nash equilibrium iff for all other possiblestrategies σ′:

U(σ, σ) ≥ U(σ′, σ) .

It is strict if the inequality is strict for all σ′ 6= σ.

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Game Theory Signaling games Population Games

Examples

Prisoner’s Dilemma

U =

(2 0

3 1

)symmetric ne: 〈0, 1〉

Hawk & Dove

U =

(1 7

2 3

)symmetric ne: 〈.8, .2〉

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Game Theory Signaling games Population Games

Symmetrizing asymmetric gamesExample: signaling game

• big population of agents• every agent might be sender or receiver• an agent’s strategy is a pair 〈s, r〉 of pure sender and receiver strategies• utilities are defined as the average of sender and receiver role:

U(〈s, r〉 ,⟨s′, r′

⟩) = 1/2(US(s, r′) + UR(s′, r)))

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Game Theory Signaling games Population Games

Example (Symmetrized 2-2-2 Lewis game)s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16

s1 〈m1, m1, a1, a1〉 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5s2 〈m1, m1, a1, a2〉 .5 .5 .5 .5 .75 .75 .75 .75 .25 .25 .25 .25 .5 .5 .5 .5s3 〈m1, m1, a2, a1〉 .5 .5 .5 .5 .25 .25 .25 .25 .75 .75 .75 .75 .5 .5 .5 .5s4 〈m1, m1, a2, a2〉 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5s5 〈m1, m2, a1, a1〉 .5 .75 .25 .5 .5 .75 .25 .5 .5 .75 .25 .5 .5 .75 .25 .5s6 〈m1, m2, a1, a2〉 .5 .75 .25 .5 .75 1 .5 .75 .25 .5 0 .25 .5 .75 .25 .5s7 〈m1, m2, a2, a1〉 .5 .75 .25 .5 .25 .5 0 .25 .75 1 .5 .75 .5 .75 .25 .5s8 〈m1, m2, a2, a2〉 .5 .75 .25 .5 .5 .75 .25 .5 .5 .75 .25 .5 .5 .75 .25 .5s9 〈m2, m1, a1, a1〉 .5 .25 .75 .5 .5 .25 .75 .5 .5 .25 .75 .5 .5 .25 .75 .5

s10 〈m2, m1, a1, a2〉 .5 .25 .75 .5 .75 .5 1 .75 .25 0 .5 .25 .5 .25 .75 .5s11 〈m2, m1, a2, a1〉 .5 .25 .75 .5 .25 0 .5 .25 .75 .5 1 .75 .5 .25 .75 .5s12 〈m2, m1, a2, a2〉 .5 .25 .75 .5 .5 .25 .75 .5 .5 .25 .75 .5 .5 .25 .75 .5s13 〈m2, m2, a1, a1〉 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5s14 〈m2, m2, a1, a2〉 .5 .5 .5 .5 .75 .75 .75 .75 .25 .25 .25 .5 .5 .5 .5 .5s15 〈m2, m2, a2, a1〉 .5 .5 .5 .5 .25 .25 .25 .25 .75 .75 .75 .75 .5 .5 .5 .5s16 〈m2, m2, a2, a2〉 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5

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non-strict symmetric ne, strict symmetric ne

Game Theory Signaling games Population Games

All pairs of sender-receiver pure strategies for the 2-2-2 Lewis game

13

9

5

1

14

10

6

2

15

11

7

3

16

12

8

4

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Reading for Next Class

Brian Skyrms (2010) “Information” Chapter 3 of “Signals” OUP.

References

Lewis, David (1969). Convention. A Philosophical Study. Cambridge, MA: HarvardUniversity Press.